1. 1
   Let $u = 3^{x}$. Equations that contain both $9^{x}$ and $3^{x}$ can be rewritten as quadratic equations in $u$. Consider the equation $9^{x} - 10 \times 3^{x} + 9 = 0$.

   (a)

   Which equation in $u$ is equivalent to $9^{x} - 10 \times 3^{x} + 9 = 0$?

   [1 mark]

   • A$u^{2} - 10u + 9 = 0$
   • B$u^{3} - 10u + 9 = 0$
   • C$9u - 10u + 9 = 0$
   • D$3u - 10u + 9 = 0$

   (b)

   Find all the solutions of $9^{x} - 10 \times 3^{x} + 9 = 0$.

   [1 mark]

   • A$x = 1$ or $x = 9$
   • B$x = 0$ or $x = 2$
   • C$x = 2$ only
   • D$x = 0$ or $x = 3$

   (c)

   Show that the equation $9^{x} + 3^{x} - 6 = 0$ has exactly one real solution, and find this solution in the form $\log_{3} a$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A graphic display calculator (GDC) may be used in this question. Consider the equation $e^{x} = x + 2$, which has no analytic method of solution.

   (a)

   How many real solutions does the equation $e^{x} = x + 2$ have?

   [1 mark]

   • A$0$
   • B$1$
   • C$2$
   • D$3$

   (b)

   Use your GDC to find the positive solution of the equation, correct to 3 significant figures.

   [1 mark]

   • A$3.15$
   • B$-1.84$
   • C$1.00$
   • D$1.15$

   (c)

   Hence solve the inequality $e^{x} > x + 2$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A graphic display calculator (GDC) may be used in this question. The population of town A, $P_{A}$ thousand, $t$ years after 1 January 2020 is modelled by $P_{A}(t) = 40e^{0.02t}$. The population of town B, $P_{B}$ thousand, is modelled by $P_{B}(t) = 65 - 0.5t$, for $t \ge 0$.

   (a)

   Find the value of $t$ at which the two towns have the same population, and find this population.

   [3 marks]

   (b)

   (i) Find the calendar year during which the population of town A first exceeds the population of town B. Justify your answer.
   (ii) Find the value of $t$ at which the model for town B predicts a population of zero, and hence comment on the validity of this model.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The functions $f$ and $g$ are defined by $f(x) = e^{2x}$ and $g(x) = 6e^{x} - 8$, for $x \in \mathbb{R}$.

   (a)

   Find the exact coordinates of the points where the graphs of $y = f(x)$ and $y = g(x)$ intersect.

   [6 marks]

   (b)

   (i) Find the exact set of values of $x$ for which $f(x) < g(x)$.
   (ii) For $k < 8$, find the value of $k$ for which the equation $f(x) = g(x) + k$ has exactly one solution. Justify your answer.

   [6 marks]

   Total for question 4: 12 marks

End of questions